Jökull - 01.12.1977, Page 82

function can be represented (Duff and Naytor,
1966) by an integral over 2
<þ(Y>)= Jg (P, R) «/>0 (R) daK, P in B, (3)
s
where G (P, R) is the appropriate Robin func-
tion, R = (x', y') and daK = dx'dy'. Evidently
lim G (P,R) = 8 (S—R), (4)
zl 0
where 8 (S—R) is the 2-dimensional delta func-
tion of S centered at R.
Moreover, the various derivatives of </> (P)
can also be represented by integral expressions
derived from (3). We are particularly interested
in the negative derivative with respect to z taken
at 2> that is, at z = 0. This quantity is con-
veniently expressed
~\4>
= h<k,
z i 0
(5)
where H is he integral-differential operator
h = -a.
z 4 0
/
G (P,R), P in B,
(6)
important case is obtained when d -» °°. We
can then construct the following eigenfunction
representation of H.
Let Uj (S) be the eigenfunctions and the
eigenvalues of on 2 with the Neumann
type boundary condition (2) on y, viz.,
-V"Uj = \jUj, j = 1,2.......... S in 2 (10)
and
3Uj/3n = 0, S on y. (11)
Considering now tlie case where d it is a
simle matter to show that any solution of (1)
which satisfies (2) can be represented by the
following eigenfunction expansion,
<f> (P) = 2ajUj (S) exp (-V*z), (12)
where the aJs are expansion coefficients. The
Laplacian on 2 with the condition (11) has the
representation
-V^JsXjUjíSJUj'ÍR). (13)
A little algebra based on (6) and (12) reveals
that
and the integration is with espect to R. This
is a cross surface differential operator which
generates the derivative of the harmonic func-
tion (f> (P) across the surface 2 111 terms of the
values of (f> (P) on 2-
The principal characteristics of H are imme-
diately revealed by the observation that apply-
ing H twice to </>0 (S) we obtain because of (1)
and the smoothness of </> (P)
H2</>o = \*<t> (P)
z j, 0
= -V;4>o' (?)
where
V2 = 3 4- 3
v 2 — zz — yy
and consequently
H = (—V22)^
(8)
(9)
that is, the operator H acts as a square root of
the 2-dimensional Laplacian.
The property (9) does not determine H uni-
quely. There is also a dependence on d. The
simplest, and in the present context the most
H = lim f^Xj^Uj (S) Uj* (R) exp (—Xj^z), (14)
z l 0 V
and has an inverse
K = H-i = lim f 2Aj-iuj (S) Uj* (R) exp (-k^z), (15)
ziOV
These results hokl for P in B, for d -» œ only,
and the integration in (13) to (15) is again witli
respect to R. Anologous results for finite deptlis
d are easily derived but some of the above
simplicity is lost. A reflection factor has to be
included in (14) and (15).
THE GRAVITY WAVE EQUATION
Let F be a layer of a homogeneous, incom-
pressible, inviscid non-rotating fluid of density
p contained in a basin equivalent to the domain
B defined above. The side walls of B are vertical
and the z-axis vertically down. Let the static
80 JÖKULL Q7. ÁR